Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index

Abstract

Let G be a simple, connected and undirected graph. The atom-bond connectivity index ($ABC\left(G\right)$) and Randić index ($R\left(G\right)$) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as $ABC-R$ index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of $ABC-R$ for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for $ABC-R$ index of such trees with given number of pendant vertices. The role of symmetry has great importance in different areas of graph theory especially in chemical graph theory.